Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network

In this paper, a bifurcation analysis is undertaken for a discrete-time Hopfield neural network with a single delay. Conditions ensuring the asymptotic stability of the null solution are found, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, fold or Neimark–Sacker bifurcations occur, but codimension 2 (fold-Neimark–Sacker, double Neimark–Sacker and resonance 1:1) bifurcations may also be present. The direction and the stability of the Neimark–Sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory.